Quantum algorithm for the Vlasov equation

The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as $O\left(N_{v}t\right)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $O[\text{polylog}\left(N_v\right)t/\delta ]$ where $\delta$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/\delta$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.

Figure 1

Implementation of $V$. Gate $W$ is given by Fig. 2. The notation of an angle $\varphi$ next to a filled dot represents the phase-shift gate, i.e., the phase $\varphi$ is applied controlled on that qubit. Our diagrams were created using the quantikz package [28].

Figure 2

Implementation of $W$. Gate $U:=U_{\text{row}}^{†}{U}_{\text{col}}$ where $U_{\text{row}}$ is given by Fig. 3 and $U_{\text{col}}$ is given by Fig. 4.

Figure 3

Implementation of ${\widehat{U}}_{\text{row}}$. The action of the rotation gate $R(·)$ is given in Eq. (24). Dashed boxes surround qubits that are inputs to a variable rotation, which does not modify those input qubits. The $a_i$ qubits are ancillas; only the behavior when they start as $|0\rangle$ is relevant.

Figure 4

Implementation of ${\widehat{U}}_{\text{col}}$. The components are described in Fig. 3.

Figure 5

Electric-field evolution in the test case. $\tilde{E}$ is the Fourier component of the electric field for wave number $k$. The envelope (envel.) is a fitted exponential, which is accurate for later times. The vanishing of the real component of $\tilde{E}$ at all times is due to our specific choice of ${\tilde{F}}_j\left(t_0\right)$.

Figure 6

Test of the simulation of the Hamiltonian in Eq. (18). The error bound is given by Eq. (34), while the actual error $\varepsilon$ is computed using Eq. (33). The failure rate is $|(1-{\widehat{P}}_g)\widehat{C}|x_g{\rangle |}^2\le 2\varepsilon$.

Figure 7

Possible preparation circuit for the state in Eq. (19). The rotation inputs $h_j$ are given in Eq. (40). The operation is repeated until the measurement on the $p$ register returns $|0\rangle$.

Figure 8

Algorithm extension for obtaining complex phase information. Gate $A$ is the original algorithm, including state preparation but not measurement. The final-state amplitude for all qubits equal to zero is modified by the phase $\zeta$ as given in Eq. (43).